Phase angles

The expressions given above assume the angle of the sine wave is zero at t = 0

If this is not the case the expression is modified by adding the angle at t = 0

Phase difference

Two waveforms of the same frequency may have a constant

phase difference.

We say that one is phase-shifted with respect to the other

Example

- see Example 2.2 in the course text.

Determine the equation of the following voltage signal.

From the diagram:

Period is 50ms = 0.05 s

Thus f = 1/T =1/0.05 = 20 Hz

Peak voltage is 10 V

Therefore, v=V_p sin 2πft  =  10sin 2π20t  =  10sin 126t

r.m.s. value of a sine wave

the instantaneous power (p) in a resistor is given by

therefore the average power is given by

where is the mean-square voltage

While the mean-square voltage is useful, more often we use the square root of this quantity, namely the root-mean-square voltage V_rms

where V_rms =

We can also define I_rms =

it is relatively easy to show that (see text for analysis)

Average value of a sine wave

Average value over one (or more) cycles is clearly zero.

However, it is often useful to know the average magnitude of the waveform independent of its polarity

we can think of this asthe average value over half a cycle…

… or as the average valueof the rectified signal

Peak factor

for any waveform the peak factor is defined as

for a sine wave this gives

Square Waves

Frequency, period, peak value and peak-to-peak value have the same meaning for all repetitive waveforms

r.m.s. value of a sine wave (cont.d)

r.m.s. values are useful because their relationship to average power is similar to the corresponding DC values

Form factor

for any waveform the form factor is defined as

for a sine wave this gives

Average and r.m.s. values

the average value of a symmetrical waveform is its average value over the positive half-cycle

thus the average value of a symmetrical square wave is equal to its peak value

similarly, since the instantaneous value of a square wave is either its peak positive or peak negative value, the square of this is the peak value squared, and

Form factor and peak factor

from the earlier definitions, for a square wave

Phase angle

we can divide the period into 360o or 2π radians

useful in defining phase relationship between signals

in the waveforms shown here, B lags A by 90o

we could alternatively give the time delay of one with respect to the other

Pause for Thought

What kind of voltage is coming from the mains supply, d.c. or a.c.?

What is the frequency of the mains supply?

What is the frequency of d.c. voltage?

What is the peak value of the mains supply?

Worked Example 2

An alternating voltage is given by v = 75 sin (200t - 2.5) volts.

Find the amplitude Vmax, the peak to peak value (2×V max), the rms value Vrms, the periodic time P, the frequency ƒ and the relative phase angle Φ

Worked Example 1

An alternating voltage is given by v = 282.8 sin 314t volts. Find the rms voltage, the frequency and the instantaneous value when t = 4 ms.

Loading effects – voltage measurement

our measuring instrument will have an effective resistance (RM)

when measuring voltage we connect a resistance in parallel with the component concerned which changes the resistance in the circuit and therefore changes the voltage we are trying to measure

this effect is known as loading

Measuring voltage and current in a circuit

when measuring voltage we connect across the component

when measuring current we connect in series with the component

Analogue Ammeters and Voltmeters

Most modern analogue ammeters are based on moving-coil meters

see Chapter 13 of textbook

Meters are characterised by their full-scale deflection (f.s.d.) and their effective resistance (RM)

typical meters produce a f.s.d. for a current of 50 μA – 1 mA

typical meters have an RM between a few ohms and a few kilohms

Loading effects – current measurement

our measuring instrument will have an effective resistance (RM)

when measuring current we connect a resistance in series with the component concerned which again changes the resistance in the circuit and therefore changes the current we are trying to measure

this is again a loading effect

Measuring direct voltages using a moving coil meter

use a series resistor to adjust sensitivity

see Example 2.6 in the set text for numerical calculations

Measuring direct currents using a moving coil meter

use a shunt resistor to adjust sensitivity

see Example 2.5 in the set text for numerical calculations

Analogue multimeters

General purpose instruments use a combination of switches and resistors to give a number of voltage and current ranges.

- a rectifier allows the measurement of AC voltage and currents

- additional circuitry permits resistance measurement

- very versatile but relatively low input resistance on voltage ranges produces considerable loading in some situations

Measuring alternating quantities

Moving coil meters respond to both positive and negative voltages, each producing deflections in opposite directions.

- a symmetrical alternating waveform will produce zero deflection (the mean value of the waveform).

- therefore we use a rectifier to produce a unidirectional signal.

- meter then displays the average value of the waveform.

- meters are often calibrated to directly display r.m.s. of sine waves.

all readings are multiplied by 1.11 – the form factor for a sine wave

- as a result waveforms of other forms will give incorrect readings.

for example when measuring a square wave (for which the form factor is 1.0, the meter will read 11% too high)

Measurement of voltage, current and resistance is achieved using appropriate circuits to produce a voltage proportional to the quantity to be measured .

in simple DMMs alternating signals are rectified as in analogue multimeters to give its average value which is multiplied by 1.11 to directly display the r.m.s. value of sine waves

more sophisticated devices use a true r.m.s. converter which accurately produced a voltage proportional to the r.m.s. value of an input waveform

Digital Multimeters

Digital multimeters (DMMs) are often (inaccurately) referred to as digital voltmeters or DVMs.

at their heart is an analogue-to-digital converter (ADC) .

- therefore we use a rectifier to produce a unidirectional signal.

Digital oscilloscope

Digital oscilloscopes use an analogue-to-digital converter (ADC) and appropriate processing.

A typical digital oscilloscope

Oscilloscopes

An oscilloscope displays voltage waveforms

A typical analogue oscilloscope

Further Study

The Further Study section at the end of Chapter 2 looks at the measurement of different forms of alternating waveform.

Have a look at the problem and then watch the video to see how you did.

🎥 Watch Video

Key Points

The magnitude of an alternating waveform can be described by its peak, peak-to-peak, average or r.m.s. value

The root-mean-square value of a waveform is the value that will produce the same power as an equivalent direct quantity

Simple analogue ammeter and voltmeters are based on moving coil meters

Digital multimeters are easy to use and offer high accuracy

Oscilloscopes display the waveform of a signal and allow quantities such as phase to be measured.

Measurement of phase difference